{ "id": "1804.05043", "version": "v1", "published": "2018-04-13T17:17:46.000Z", "updated": "2018-04-13T17:17:46.000Z", "title": "Representations of reductive groups over finite local rings of length two", "authors": [ "Alexander Stasinski", "Andrea Vera-Gajardo" ], "comment": "15 pages", "categories": [ "math.RT", "math.GR" ], "abstract": "Let $\\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $W_{2}(\\mathbb{F}_{q})$ be the ring of Witt vectors of length two over $\\mathbb{F}_{q}$. We prove that for any reductive group scheme $\\mathbb{G}$ over $\\mathbb{Z}$ such that $p$ is very good for $\\mathbb{G}\\times\\mathbb{F}_{q}$, the groups $\\mathbb{G}(\\mathbb{F}_{q}[t]/t^{2})$ and $\\mathbb{G}(W_{2}(\\mathbb{F}_{q}))$ have the same number of irreducible representations of dimension $d$, for each $d$. Equivalently, there exists an isomorphism of group algebras $\\mathbb{C}[\\mathbb{G}(\\mathbb{F}_{q}[t]/t^{2})]\\cong\\mathbb{C}[\\mathbb{G}(W_{2}(\\mathbb{F}_{q}))]$.", "revisions": [ { "version": "v1", "updated": "2018-04-13T17:17:46.000Z" } ], "analyses": { "keywords": [ "finite local rings", "witt vectors", "finite field", "group algebras", "reductive group scheme" ], "note": { "typesetting": "TeX", "pages": 15, "language": "en", "license": "arXiv", "status": "editable" } } }